Find the sum of n terms of the series 1+3 +7 +15 + …

Options

A. `[2^n -1)-n]`
B. `[(2^n -1) + n]`
C. `[2(2^n -1)-n]`
D. none of these

Answer

To find the sum of the first n terms of the series 1 + 3 + 7 + 15 + ..., we first need to identify the pattern in the series.

### Step 1: Identify the Pattern
The given series is:
- 1 (1st term)
- 3 (2nd term)
- 7 (3rd term)
- 15 (4th term)

Let's observe the differences between consecutive terms:
- 3 - 1 = 2
- 7 - 3 = 4
- 15 - 7 = 8

The differences are 2, 4, and 8, which are powers of 2:
- 2 = 2^1
- 4 = 2^2
- 8 = 2^3

This suggests that the nth term can be expressed in terms of powers of 2.

### Step 2: Find the nth Term
We can observe that the nth term can be represented as:
- T(n) = 2^n - 1

To verify:
- For n = 1: T(1) = 2^1 - 1 = 1
- For n = 2: T(2) = 2^2 - 1 = 3
- For n = 3: T(3) = 2^3 - 1 = 7
- For n = 4: T(4) = 2^4 - 1 = 15

### Step 3: Sum of n Terms
The sum of the first n terms, S(n), can be calculated as:
\[ S(n) = T(1) + T(2) + T(3) + ... + T(n) \]
\[ S(n) = (2^1 - 1) + (2^2 - 1) + (2^3 - 1) + ... + (2^n - 1) \]

This simplifies to:
\[ S(n) = (2^1 + 2^2 + 2^3 + ... + 2^n) - n \]

### Step 4: Use the Formula for the Sum of a Geometric Series
The sum of a geometric series can be calculated using the formula:
\[ \text{Sum} = a \frac{(r^n - 1)}{(r - 1)} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In our case:
- \( a = 2 \)
- \( r = 2 \)
- \( n = n \)

Thus, we have:
\[ S(n) = 2 \frac{(2^n - 1)}{(2 - 1)} - n \]
\[ S(n) = 2(2^n - 1) - n \]
\[ S(n) = 2^{n+1} - 2 - n \]

### Final Answer
The sum of the first n terms of the series is:
\[ S(n) = 2^{n+1} - n - 2 \]